Time Domain Analysis of Discrete Systems

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Time Domain Analysis of Discrete Systems

• Difference Equations
• Chapter 9
• Lecture 6

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Difference Equations

• Last time we looked at 3 discrete systems and
  produced Difference Equations.
• Similar to Differential Equations for
  cont-systems
• x(t)

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• General Form
• a0yn a1yn-1 a2yn-2 aNyn-N
• b0xn b1xn-1 b2xn-2 bMxn-M
• Where all as and bs are constants
• Must solve for yn
• Easiest solution method Iteration
• ex yn 0.5yn-1 xn n 0 1
  2 3
• xn un xn
• y-1 -2 0.5yn-1
• yn

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Solving Difference Equations

• 2 parts Zero Input Zero State
• yn yCn
  yPn
• homogeneous forced
• complementary
  particular
• natural
• A continuous example differential equation

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Back to Solving Difference Equations

• Recall yn yCn yPn
• Start with yCn
• Zero input xn 0
• Assume a general form
• And if
• a0yn a1yn-1 a2yn-2 aNyn-N

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• Plugging in
• a0Czn a1Cznz-1 a2Cznz-2 . aNCznz-N 0

How many cases could we have? ____
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• Ex 1a yn yn-1 2yn-2 xn
• y-1 -2.5 and y-2 1¾
• xn 0 ( since we are only finding ycn )
• Lets put this into Advanced Operator Form
• Find Characteristic Equation

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• Ex 2a yn 4yn-1 4yn-2 xn
• y-1 -0.5 and y-2 0
• xn 0 ( since we are only finding ycn )
• Find Characteristic Equation

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• Ex 3a yn yn-1 1.25yn-2 xn
• y0 1 and y-1 0
• xn 0 ( since we are only finding ycn )
• Find Characteristic Equation

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Complete Solutions to difference Eq

• Recall yn
• Lets work on yPn
• ALWAYS looks like the input signal
• xn 0.9n ypn
• xn n2 ypn
• xn cos(2n) ypn

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• Ex 1b yn yn-1 2yn-2 4un

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• Ex 2b yn 4yn-1 4yn-2 xn, let
  xn (1/2)nun

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Impulse Response

• know yn
• ypn input impulse
• so, yn hn ?

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Impulse Response, cont

• Find hn for yn -1/2yn-1 xn
• Recall dn

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